Previous Attempts to Measure the Sound-Intensity Vector
The sound-intensity vector, or sound power flow per unit area, is defined as the product of sound pressure and sound velocity. It is difficult to measure as a function of time and is usually determined as a function of frequency. Ways of measuring sound intensity are described in    1. F. J. Fahy, 1995, “Sound Intensity”, Second Edition, E& FN Spon, An imprint of Chapman and Hall, London.    2. Anon., 1996, “Instruments for Measurement of Sound Intensity”, Standard ANSI S1.9-1996, American National Standards Inst.Sound intensity is not measured directly. It involves the use of a measurement calculation. Generally only one component of the intensity vector is measured using a pair of microphones. Two-microphone instruments are discussed almost exclusively by Fahy and in the ANSI standard, except that on pages 112 and 113 of Fahy two instruments are described that measure all three components of the intensity vector using four or more microphones.
The first of these instruments uses a probe consisting of three pairs of condenser microphones aligned face-to-face along three Cartesian axes. A probe of this type is manufactured, for example, by GRAS Sound & Vibration ApS in Denmark, as model number 50VI, and is described in    3. P. Rasmussen, 1989, “Source Location using Vector Intensity Measurements”, Sound and Vibration Magazine, Vol. 23, pages 28–33.Each pair of microphones is phase matched to provide better accuracy at low frequencies. Six microphones are used because it is easier to select phase matched condenser microphones in pairs, during manufacture. However six is more than is necessary, making the probe particularly unwieldy. A requirement for accuracy in the measurement calculation is that the sensitivity of the probe is omnidirectional; i.e. is equally sensitive to sound from all directions. However, because the six microphones have manufacturing variability and do not respond identically, the sensitivity of the GRAS probe is not omnidirectional. The usual calibration in the field at a single frequency cannot correct for this. Also the structure of the six-microphone probe prevents it from making measurements close to a source. Two sizes (typically 12 and 50 mm) of solid spacers are used to separate the faces of the three pairs of microphones in the probe. The smaller spacer is used for accuracy at higher frequencies and the larger for accuracy at lower frequencies. This means that the probe can not make accurate measurements at the same measurement point, concurrently at both low and high frequencies. Although it would be possible to output data on sound pressure and velocity with this instrument, no attempt was made to do this. Also azimuth-elevation plots were not used by P. Rasmussen to represent the direction of a sound source.
The second instrument for measuring the three components of sound intensity cited by Fahy was manufactured by Ono Sokki in Japan as model MI-6420. It uses a probe consisting of four condenser microphones positioned at the vertices of an imaginary regular tetrahedron. The tetrahedral arrangement is well known. Originally it appears to have been mentioned by    4. G. Rasmussen, 1985, “Measurement of Vector Fields”, pages 52–58, Proc. Second International Congress on Acoustic Intensity, CETIM, Senlis, France.and discussed later by    5. L. M. C. Santos, C. C. Rodrigues and J. L. Bento-Coelho, 1989, “Measuring the Three Dimensional Acoustic Intensity Vector with a Four-Microphone Probe”, Proceedings of INTER-NOISE 89, 965–968.A regular tetrahedron has the basic geometric property that lines joining the midpoints of opposite edges form a set of Cartesian axes, thus providing a ready-made coordinate system for determining the components of the sound intensity vector, with the measurement point at the origin. This property is used by the Rasmussens and by Santos et al, but surprisingly it was not used in the Ono-Sokki instrument. Instead one of the coordinate axes is assumed to pass through a microphone that protrudes ahead of the other three, as described by    6. H. Suzuki, “Three dimensional acoustic intensity measuring device”, Japanese Patent No. 0528898, Nov. 2, 1993.This coordinate system appears to have been used first by    7. K. Segiguchi, S. Kimura and T. Hanyuu, 1992, “Analysis of Sound Field on Spatial Information using a Four-Channel Microphone System on Regular Tetrahedron Peak Point Method”, Applied Acoustics, 37, 305–323.In this latter paper the direction of a sound source is determined using time of flight, rather than from the direction of the sound-intensity vector. An account of the measurement calculation for the Ono-Sokki instrument is given in    8. H. Suzuki, S. Oguro, M. Anzi and T. Ono, 1995, “Performance evaluation of a three dimensional intensity probe”, Journal. of the Acoustical Society of Japan, (E), 16, 4, pages 232–238.Because of the singular nature of the coordinate system, the calculation is complicated and subject to error. Even though the Suzuki probe uses fewer microphones than the GRAS probe it is still somewhat unwieldy and cannot detect sound well from the direction of the microphone preamplifiers and leads. As with the GRAS instrument, the microphones in the probe do not respond identically, so that the sensitivity of the probe is not omnidirectional. Also the instrument cannot make accurate measurements concurrently at both low and high frequencies. No attempt is made to present data on sound pressure and sound velocity. Also the Ono-Sokki and the Segiguchi instruments do not use practical azimuth-elevation plots to indicate the direction of a sound source.Condenser and Electret Microphones
The GRAS, Ono-Sokki and Segiguchi instruments all use condenser microphones. A condenser microphone is generally made with a stainless steel membrane and other metal parts. It is very stable, making it suitable for use as a standard. However, it requires a pre-amplifier, usually contained in a metal tube, which has to be included as part of the structure of the probe Condenser microphones are relatively large. The microphones in the GRAS and Ono-Sokki systems are about 12.7 and 7 mm in diameter. The center of sensitivity of a microphone can not be assumed to be the geometric center of the membrane. Hence, for larger microphones, it is not possible to make an accurate determination of the inter-microphone spacing in the probe. This spacing has to be known accurately for best accuracy in the measurement calculation. As shown later, the problem of locating the center of sensitivity of a microphone is greatly alleviated with small microphones that use a thin film of polarized material called an electret
Measurement Calculation
Sound intensity is generally determined using the cross-spectral formulation, first derived in 1977 (see Fahy) for a two-microphone probe. This formulation relates sound intensity to the cross spectrum of the sound pressures at the microphones. It was discussed by    9. J. Y. Chung, “Sound Intensity Meter”, U.S. Pat. No. 4,236,040, Nov. 25, 1980.The formulation uses finite-difference approximations based on the requirement that the microphone spacing is less than the wavelength of the sound being measured. This requirement places an upper limit on the frequency range of the measurement. A three-dimensional version of the cross-spectral formulation was used by    10. R. Hickling and W. Wei, 2000, “Use of pitch-azimuth plots in determining the direction of a noise source in water with a vector sound-intensity probe”, Journ. Acoust. Soc Amer, 97(2), pp 856–866.In this paper the probe consists of four hydrophones in the tetrahedral arrangement. Since the hydrophones do not have the same frequency response, the sensitivity of the probe is not omnidirectional. Also the hydrophone spacing is not known precisely so that the measurement calculation is correspondingly less accurate and the measurement point is not known precisely. Hickling et al employ azimuth-elevation plots to represent the direction of a sound source. Other types of representation are used by P. Rasmussen and by Segiguchi et al.